Weird invisible face issue
In this case, you have a simple problem of missing faces. One way to verify this is to install the MeshLint addon for Blender. Then use the "Check Non-manifold Elements" tool. This will find any holes in your mesh. 

To fix the holes, select the edges around one hole and press F. And repeat for any remaining holes.



Other common mesh problems can be fixed by removing vertices that share the same space. Called doubles. To do so, press W and select Remove Doubles.